The UV sensitivity of the Higgs potential in Gauge-Higgs Unification

In this paper, we discuss the UV sensitivity of the Higgs effective potential in a Gauge-Higgs Unification (GHU) model. We consider an $SU(\mathcal N)$ GHU on $\mathbf M^4\times S^1$ spacetime with a massless Dirac fermion. In this model, we evaluate the four-Fermi diagrams at the two-loop level and find them to be logarithmically divergent in the dimensional regularization scheme. Moreover, we confirm that their counter terms contribute to the Higgs effective potential at the four-loop level. This result means that the Higgs effective potential in the GHU depends on UV theories as well as in other non-renormalizable theories.


I. INTRODUCTION
The standard model (SM) of particle physics is a Yang-Mills theory symmetric under SU (3) c × SU (2) L × U (1) Y gauge transformations. In the SM, the gauge symmetry is spontaneously broken via the Higgs mechanism, which is caused by the nonzero vacuum expectation value (VEV) of the Higgs boson. As a result, physical quantities such as the mass of a particle include the Higgs VEV. By measuring the physical parameters including the Higgs VEV, the SM is confirmed to be consistent with phenomena at Large Hadron Collider [1,2].
While the phenomenology below the electroweak (EW) scale is understandable by the SM, it is concerned that the SM has difficulty explaining the scale hierarchy between the EW and UV theories such as the grand unified theory (GUT) or quantum gravity because of the dangerous quadratic divergences derived from the Higgs boson. In a model with supersymmetry (SUSY) [3][4][5][6][7], there is a superpartner for each particle to cancel the quadratic divergences consequently. Instead of the SUSY scenarios, we can invoke gauge symmetry in a non-SUSY theory defined on extra-dimensional spacetime to protect the Higgs mass term.
In the GHU models, the Higgs bosons are identified with the Yang-Mills Aharonov-Bohm (AB) phases. Therefore, the Higgs boson has no potential at the tree level. Meanwhile, at the loop level, the Higgs potential is generated by the AB effect due to the non-simplicity of spacetime. This symmetry breaking mechanism is called the Hosotani mechanism [11,13].
In the previous work [18], we confirmed that the Higgs potential does not suffer from the divergence at the two-loop level in a non-Abelian gauge theory defined on M 4 × S 1 spacetime, where M n is the n-dimensional Minkowski spacetime with n ≥ 1 and S 1 is a circle. Besides, we proceeded to discuss the finiteness of the Higgs potential on M 5 × S 1 spacetime, which is related to this paper. Evaluating the four-Fermi diagrams at the one-loop level, we obtained the logarithmic divergences contributing to the Higgs potential. Hence, the Higgs potential is UV sensitive on the six-dimensional spacetime. This is consistent with the non-renormalizability of higher-dimensional gauge theory. Based on this result, we concluded that the Higgs potential would be also suffering from the divergence in the five-dimensional spacetime.
In this paper, we go back to M 4 × S 1 spacetime again and explicitly show that the Higgs potential depends on UV theories in an SU (N ) GHU model. Since there are no logarithmic divergences at the one-loop level in odd-dimensional theories, we consider divergences at the two-loop level in the dimensional regularization scheme. The four-Fermi diagrams, in practice, are evaluated at the two-loop level. We find that they are indeed logarithmically divergent and their counter terms contribute to the Higgs potential at the four-loop level.
This fact means the UV sensitivity of the Higgs potential in this GHU model. Since we use a simple setup, the Higgs potential would generically be UV sensitive in other GHU models.
The remainder of this paper is organized as followes. In Sect. II, we briefly describe the Hosotani mechanism along with the theoretical setup. In Sect. III, we explain how to evaluate the divergence of the Higgs potential and show that the Higgs potential receives a contribution from counter terms to the divergences that cannot be subtracted by the renormalization to the gauge coupling. Finally, we summarize the results obtained in this paper in Sect. IV.

II. HOSOTANI MECHANISM
In this section, we describe a theoretical setup used in this paper and review the Hosotani mechanism.
Since the Hosotani mechanism is a quantum effect related to the global structure of spacetime, let us consider M 4 × S 1 as an example of non-simply connected spacetime, where M 4 and S 1 are the four-dimensional Minkowski spacetime and a circle with radius R respectively. We use coordinates x µ with µ ∈ {0, 1, 2, 3} for M 4 and y ∈ [0, 2πR) for S 1 . As mentioned above, on a spacetime with a hole, the fifth component of the gauge boson, A a 5 , has its VEV expressed by where θ a 's are the Yang-Mills AB phases around S 1 and g is the coupling constant. Here, a denotes the group index.
Through this paper, we use the background field method [23] for calculating the contri-butions to the Higgs effective potential and A a 5 's are shifted by its VEV; . ( Due to compactified extra-dimension, the boundary conditions on field functions are introduced. We consider a massless Dirac fermion, ψ, as only one type of matter field and suppose A a M and ψ satisfy ψ(x µ , y + 2πR) = e iβ ψ(x µ , y), where M ∈ {0, 1, 2, 3, 5} and β ∈ [0, 2π).
The Lagrangian we consider with an SU (N ) gauge symmetry is where the gauge fixing terms, L GF , and the Faddeev-Popov ghost terms, L ghost , are given by where n ∈ Z denotes the Kaluza-Klein (KK) modes. To shift the fifth component of a momentum, we consider the following gauge transformations; ψ (x µ , y) → e −i θ a τ a 2πR y ψ (x µ , y), where A M = A a M T a . Without any boundary conditions, arbitrary θ a can be gauged away by the above gauge transformations. With Eqs.
where I is the identity matrix. Remaining θ a 's become physical degrees of freedom. Due to the gauge symmetry, θ a has no potential at the tree level. Under the boundary conditions which characterize the non-simplicity of spacetime, however, its potential is generated by loop corrections as shown in [11,[13][14][15][16][17][18][19]. When the minimum point of the potential is nonzero, the gauge symmetry is dynamically broken and the gauge bosons become massive. This is how the Hosotani mechanism works.

III. HIGGS POTENTIAL DIVERGENCE AT THE FOUR-LOOP LEVEL
Up to the two-loop level, we have found no divergence of the Higgs potential in the previous works [11,[13][14][15][16][17][18][19]. However, since the higher-dimensional gauge theory is non- , which is logarithmically divergent. To obtain its divergent part, we concentrate on loop momenta going around the UV region. Ignoring momenta lying external lines, we have div = 1 2πR where α and γ are spin indices ofψ, and β and δ are those of ψ. i, j, k, l represent indices of τ a 's. Note that all summation indices in this paper run all integers. In the previous work [18], we derived the following formula 1 ; where Θ is an arbitrary Hermitian matrix. Here, S(·) denotes an analytic function and its generalization to a matrix-valued one. Using Eq. (15) for Θ = 0, we get div = −ig 6 Let us define I by where x and y are space-like vectors. For the spacetime dimension D = 5 − 2 with > 0, I is calculated using a formula deduced in Appendix A. Integrals in Eq. (16) can be rewritten as a derivative of I; The behavior of the integrand in the UV region is shown in Eq. (A18); after integration over k and angular variables, the remaining (radial) integral has the form, where a, b, r and β are independent of x and y. Here, K r (z) is the modified Bessel function of the second kind and 0 F 1 (a; z) is a generalized hypergeometric function. Note that 0 F 1 (a; z) is expressed by the Bessel function of the first kind, J α (z); Plugging Eq. (20) into Eq. (19), we see that the integrand is a multiplication of J α , K r , and the power of |p E |. Therefore, the UV divergence of I is suppressed by K r for its exponential dumping when To evaluate the UV divergence of I, we set m 1 = m 2 = 0. Substituting Eq. (A20) into Eq. (18), we obtain . (21) The derivatives are given by In Appendix A, it is shown that Using this formula, we have Repeating the above procedure for the other two-loop four-Fermi diagrams, we find the -poles at the two-loop level. They are shown explicitly in Appendix B. The divergence has the form, where X denotes diagrams with four fermion legs and C X is a constant. Here, G (1,2) X and T (1,2) X are products of γ M 's and τ a 's respectively. The above example corresponds to T (1) The following counter term is introduced to cancel the above divergence 2 ; where δ 4F = δ div 4F + δ fin 4F . Here, δ fin 4F is an arbitrary constant and δ div 4F is defined as to subtract the -pole.
Closing the fermion lines, we get a contribution to the Higgs potential from L CT with nontrivial θ-dependence; where we have traced out matrix indices of both γ M 's and τ a 's. Using Eq. (15), we get In the previous work [18], we have derived that From this formula, we obtain The contributions from each diagram are explicitly written down in Appendix B. To get V CT (θ) in an Abelian gauge theory, we replace τ a 's with Q, the U (1)-charge of ψ. We have computed V CT (θ) in an SU (2) gauge theory with a fermion in the fundamental representation and an Abelian case with a fermion having the U (1)-charge Q = 1, which is shown in FIG. 1. Therefore, it is concluded that the θ-dependent part of the Higgs potential is UV sensitive. V CT (θ) in an Abelian gauge theory with a fermion whose U (1)-charge equals to one. β is specified to be zero. In the previous work [18], it was shown that contributions to the Higgs potential from the one-loop four-Fermi diagrams vanished in the Abelian gauge theory. Based on this numerical calculation, we reject the all-order finiteness of the Higgs potential in an Abelian gauge theory.

IV. SUMMARY
In this paper, we investigate the finiteness of the Higgs potential beyond the two-loop level in the GHU by evaluating the loop corrections explicitly. While the Higgs potential was found finite at the one-or two-loop levels on many non-simply connected manifolds, its finiteness at higher-order had been unclear. As suggested in the previous work [18], it is shown that the Higgs potential receives the nontrivial θ-dependent contributions from the counter terms for the four-Fermi diagrams on M 4 × S 1 at the four-loop level and, thus, it is UV sensitive.
For logarithmic divergences found in this paper, when we impose a UV cutoff on the GHU to make sense as an effective field theory, the maximum value of the cutoff, denoted as Λ max , satisfies ln(RΛ max ) ∼ g 2 Λ max . Hence, in the GHU, the perturbation is valid at most around the compactification scale.

ACKNOWLEDGMENTS
The author A.Y. thanks Junji Hisano for his tremendous supports and meaningful discussions and also thanks Yutaro Shoji for his advice and unconditional dedication to improving this paper.

Appendix A: Two-loop integrals
This appendix is dedicated to evaluating a following integral; where s, t, u are positive constants satisfying s + t + u > D and x, y are space-like vectors independent of p and k.
Introducing the Feynman parameters, we get In the previous work [18], we showed where Re(s) > 0, p 2 + m 2 = 0. Here, K r (z) is the modified Bessel function of the second kind. Using this formula, we have where we have scaled β to (1 − α)β.
Using the Wick rotation, we get Carrying out the integral over all angles except θ, the angle between p E = (p 0 E , p E ) and where Ω D is the area of the unit sphere in the D-dimensional space; The integral over θ is evaluated as where x 0 E ≡ x 0 and y 0 E ≡ y 0 . We evaluate angle integrals with a similar way to Eqs. (A10) and (A12); for Re(a) > 0 and b ∈ R.
Applying the above formulae to F, we obtain regardless of whether x − βy is space-like or time-like. Therefore, after integration over k and p, we get By expanding the last factor, finding the -pole of I comes down to calculating the following integral; where σ, τ , κ, and λ are arbitrary constants.
Setting γ to be we have where B(x, y) is the beta function; Appendix B: Table of divergence and its contribution to the Higgs potential As shown in Sect. III, at the two-loop level, the divergences of the four-Fermi diagrams have the form, where G (1,2) X and T (1,2) X are products of γ M 's and τ a 's respectively. Here, C X is a constant and X denotes divergent diagrams without crossing of the fermion lines. The second term represents the same diagram with the fermion lines being crossed.